Engineer&#39;s slide-rule.



110,767,170. EATENTED AUG. 9, 1904.

y L. w. vEDSEETEAL. ENGINEEEfs 'SLIDE RULE. APPLICATION FILED MAB. 4,1904. No MoDEL.

UNITED STATES Patented August 9, 1904.

PATENT EEICE.

ENGINEERS SLIDE-RULE.

SPECIFICATION forming part of Letters Patent No.-'76'7,170, dated August9, 1904,

Application filed March 4, 1904. Serial No. 196,581. (No model.)

To all 1,072,071?, t may concern:

Be it known that I, LEON WTALTER ROSEN- 'rHAL, a citizen ofthe UnitedStates, residing in New York, borough of Manhattamand State of New York,have invented certain new and useful Improvements in EngineersSlide-Rules, of which the following is a specification.

In the well-known Mannheim slide-rules in general use it was impossibleto find the product of more than two numbers at one setting. To enableengineers to find the product of three numbers at one setting, a rulehas been proposed, known as the duplex rule, having scales on both sidesof the same, which, however, increase the cost of such rules, decreasetheir scope of application, and, furthermore, require the inverting ofthe entire rule in order to obtain the final reading' in each case.

My invention relates to engineers sliderules, and more particularly toimprovements on the Mannheim rules, and has for its object to provide arule whereby the product or contin ued quotient of three numbers may bemore readily and accurately determined, a solution of many common anduseful problems facilitated and r'nade possible, and .the resultsobtained with convenience and accuracy, and yet the general physicalcharacteristics of the Mannheim type of slide-rule retained.. For thispurpose the invention consists of anengineers slide-rule having twofixed bars connected with and arranged parallel to each other by across-piece the under face of which is adapted for the usual table ofconstants, one of said bars having a logarithmic scale progressing' fromthe left to the right, the other having a scale progressing from theleft to the median point of the graduated length of the rule and theother scale from this pointy toward the right of the rule, and a movableslideinterposed between these bars having' one scale in juxtapositionwith the scale of the bar progressing from the left to the right for fhereinafter as the.scale of cubes, arranged in cubical relation to thescale of one of the fixed bars of the rule and in three-half-powerrelation to the two scales of the other fixed bar; and the inventionconsists of certain novel features by the use of which many calculationsmay be greatly facilitated and additional computations made possible, aswill be more fully described hereinafter and finally pointed out in theclaims.

In the accompanying drawings, Figure l represents an upper' face view ofmy improved slide-rule. Fig. 2 also shows an upper face View of the samewith the slide entirely removed, so as to show the scale of cubes. Fig.3 shows an upper face View of the slide. Fig. 4 shows a transversesection taken on line 4 4, Fig. l. Fig. 5 is a detail view of the end ofthe slide arranged so as to facilitate the readings when the scale ofcubes is used, and

Fig. 6 is a detail view showing a modified.

forni of the end of the slide.

' Similar letters of reference indicate corresponding parts.

Referring to the drawings, A represents the main piece of my improvedslide-rule, which main piece consists of two fixed parallel bars Gr andHand a member or cross-piece K, connecting' the under faces of the fixedbars, and having a plane under face provided with a table of constantsor other frequently-occurring and useful information in the usualmanner. The bars G and H are provided at their inner edges withlongitudinal grooves 7L, in

which is guided a slide M, that is provided at its edges withlongitudinal tongues m. The slide M is guided in the longitudinaldepression of the main piece Abetween the fixed bars Gr and H. The fixedbar H is provided with a logarithmic scale D, progressing from the lefttoward the right for the graduated length of the rule, and the bar G isprovided with two logarithmic scales, the scale A/ progressing from theleft-hand end to the median point of the bar and the scale A2 from themedian point toward the right-hand end of the bar.

R represents a runner, similar-to that of the ordinary rules, providedwith a piece of glass or other transparent material, having a fine IOOdistinct line marked across the under side of its face and placedarbitrarily at the center of the glass.

The under face of the slide M is arranged with three scales, well knownin slide-rules and not shown in the drawings. One of these scales is forthe purpose of trigonometrical computations involving the sines ofangles and is ordinarily used in conjunction with the scales A A2 of thefixed bar G, known as the scale of squares. The second scale on theunder face of the slide is for the purpose of trigonometricalcomputations involving the tangents of angles and is ordinarily used in,conjunction with the scale D of the fixed bar H. The third scale on theunder face of the slide M is generally arranged intermediately betweenthe trigonometrical scales just mentioned, has equally-spaced divisions,and is ordinarily used in conjunction with the scale D for the solutionof problems in which actual mantissae of logarithmic numbers are to benoted.

The parts so far described are the same as in the well-known Mannheimslide-rule. The slide M is provided on its upper face on. one side ofits longitudinal center line with a logarithmic scale C, progressingfrom the left toward the right of the slide for the graduated length ofthe same and in juxtaposition with the scale D of the bar H; but insteadof having two additional scales on the other side of the longitudinalcenter line of the slideprogressing in the same direction and injuxtaposition to the scales of the fixed bar G, as heretofore, theimproved slide M has two logarithmic scales B B2, each progressing fromthe median point of the graduated length of the slidein oppositedirections to each other to the ends of the same. By this arrangement myimproved slide has two logarithmic scales both progressing from themedian point toward the ends of the slide, the scale B2 progressing fromthe median point toward the right-hand end of the rule and the scale Bprogressing from the median point toward the lefthand end of the rule,whereby the right-hand scale B2 is in exact coincidence 'with theright-hand scale A2 of the bar G, while the left-hand scale B is inreverse with respect to the scale A of the bar G; By this arrangementone is enabled to find the product of three numbers at one setting ofthe slide or the continued quotient of three numbers-that is, one numberdivided by the product of two numbers-likewise at one setting of theslide, manipulations which require two settings with the ordinaryMannheim rule, together with additional time and labor and decreasedaccuracy connected therewith.

In the drawings the scale B is shown to progress in opposite directionto that of scale A. Instead of this arrangement the scale B may equallywell be arranged so as to be in juxtaposition to scale A, and thedirection of the scale B2 be made opposite to that of scale A2, or theslide M may be provided with two scales progressing in the samedirection and either one of the scales constituting half the graduatedlength of the rule arranged so as to progress in opposite direction toeither of the scales of the slide. The arrangement of one of the scalesof the slide, either as shown in the drawings or else with the reversed`scale in the position there given to B2, would require the least changein the ordinary rule and would be the preferable form.

By the proposed arrangement of the scale B with respect to the scale Athe reciprocal of numbers is dealt with instead of the nurnbersthemselves. Thus by setting the lefthand index of the scale B incoincidence with the left-hand index of the scale A or A2 it is foundthat in coincidence With all numbers on the scale A or A2 will bedirectly found the respective reciprocals on the scale B. Thus thereciprocal of two is .5, of four is .25, of five is .20, &c., as may beseen by comparing scale A of Fig. 2 with scale B of Fig. 3. s

The product of two numbers is exactly equivalent to the quotientobtained by dividing either factor by the reciprocal of the other, andthe division of one number by another is exactly equivalent to theproduct obtained by multiplying the dividend by the reciprocal of thedivisor. Hence to multiply two numbers either one of the factors onscale B is placed in coincidence with the second factor on scale A orA2, and over one index of B is found the product on A or A2. Thus inFig. 1 where 4 on scale B is in coincidence with 36 on scale A theirproduct 144 is read on scale A above the left-hand index of'scale B.Similarly, to divide one index of B is brought under the dividend oneither scale A or A2 and the quotient is read on A or A2 above thedivisor on scale B. Thus to divide one hundred and forty-four by two theleft-hand index of scale B' is 'placed in coincidence with 144 on scaleA and over 2 on scale B is found the quotient 727 on scale A.

In practice it has been found easier and more accurate to multiply thanto divide, because with the Mannheim or ordinary arrangement of thescales it is necessary to set the divisor in coincidence with thedividend, nearly always requiring the use ofthe indicator-line of therunner as a guide to the setting, whereas in multiplication one index isused in the initial setting. With the arrangement of scales as formingpart of my improved sliderule the process of division becomes one ofmultiplication, thus obviating the disadvantages of the common method ofdivision.

With my improved slide-rule multiplication of three factors isaccomplished at one setting, since in using scale B the product of twofactors is found over its index, which is the re- IOO quired setting formultiplication of that product by any number on scale B2. Thus theproduct of two multiplied by seventy-two multiplied by four will befound in the following manner: Set 72on B to 2 onA and over 4 on B2 lind2 576 on A2. Similarly, in the division of one number by two others thetime and labor is much reduced, while the accuracy .of the final resultis increased. For example, the solution of thirty-six divided bytwenty-live multiplied by six would be obtained in this manner: Set 25on B2 to 36 on A2 and over 6 on B read .24 on A.

The multiplication or division of more than three factors may beperformed by the same methods with a proportionate saving of time andlabor and an increase of accuracy in leach case. Thus the product offour or live factors may be determined in two settings instead of threeor four, as with the Mannheim type of slide-rules hitherto in use.Similarly, the division of one number by three or four factors is foundin two settings instead of in three or four as heretofore.

Proportion is either direct or inverse, the latter necessitating for itsmost rapid and convenient solution with the ordinary arrangement ofscales that the slide be withdrawn from between the fixed bars andturned end for end before replacing.` This operation requires a certainamount ofV time and is liable to subject the slide to .wear and damageand in a short time render the rule unfit for accuf rate determinations`ings are not so accurate, owing to the fact that the numbers on theslide are then upside down. By the use of scale B as arranged inverseproportion is as readily solved as direct. The following problemillustrates this method: Vhat will be the speed of rotation of a pulleysix inches in diameter when driven by a belt from another pulley havinga diameter of thirty-six inches and making four hundred revolutions perminute-that is, 36 :6 :1 X: 400? Solution: Set 36 on B to "400 on A andover 6 on B lint 2,400 revolutions per minute on A.

Also in many common problems of design where the product of'two factorsis a fixed quantity all possible combinations which will give thatproduct are directly found in coin cidence on scales B and Aor A2 whenone index of B is set on the constant quantity on identical to eachother, on the face of the depression of the main piece A, intermediatelybetween the xed bars G and H. These scales are of such relation tothescales D of the bar H that any number of the bar H is in line with itscube found on scale E, E2, or E2.

Furthermore, the reada ber of the bar Gr is in line with its three-halfpower found on thescales E, E2, or E2. Thus the cube or third power oftwelve is found to be 1728 on scale E in line with 12 of the bar H.Conversely. the cube-root of twenty-seven is found to be 3 on scale D inline with 27 of scale Similarly, thel three-half power of four is foundto be 8 on E in line with 4 on scale A of bar G. Conversely, thetwo-third power of eight is found to be 4 on scale A in line with 8 ofscale E In using the scale of cubes it is found desirable to provideeach end of the slide with a recess X, extending from the extreme end ofthe slide as far as the line of the nearer index of the scales on theslide, as shown in Eig. 3 and enlarged in Fig. 5. IVhen the scale ofcubes is used in the course of solving a problem, the innermost edge ofthe recess X, in line with the index of the slide, becomes theindicating edge N. In place of a recess the slide may be providedwithaslot Y, having a transverse indicator-tlnead Z in alinement withthe index of the scale on the slide, as shown in Fig. 6, so as to obtainan accurate setting of the numbers of any other logarithmic scale on therule or slide when used in conjunction with the scale of cubes.

Vhen a slide is provided having a recess X, as shown in Fig. 5, the endsof the slide, are preferably provided with a metallic crossband T, so asto facilitate the manipulation of theV slide and render it less liableto injury. Instead of the recess X or slot Y the ends of the slide maybe left intact, in which case the edge itself of the slide would beused. ln this case allowance for the distance between the end of theslide and the index-line would have to be made for each setting', whichis accomplished by providing the runner with a short parallel line R2 oneach side of the indicator-line R of the same. The recess X or slot Ypermits, however, the direct use of constant-demarkations,` whichrepresent the fixed points corresponding to the constants in mostgeneral Lise. Thus the indicating edge N or line Z would be set on c 7)0' CZ', &c., representing values of constants and the product of thesame with any factor obtained at one setting. The constants themselvesmay be designated directly as 7T or a system of symbols as a t o CZ,&c., used, the interpretation of which would be found on the under faceof the cross-piece K.

In using the scale of cubes with all numbers having one, four, seven, orten, &c., digits scale E is used, for numbers having two, five, eight,or eleven, &c., digits scale E2 is used, and for those numbers havingthree, six, nine, or twelve, &c., digits scale E3 is used.

The following are some of the many prob- IOO 1. 32 I 32 3 3 :2 Over 3 onscaleD set 3 on B and over 3 on B2 read 81 on A2--that is, the factorthree is squared by reading' directly from scaleD to scale A and theproduct multiplied by the two other factors all at one setting.

2. \/t b c:? SetaonBtobonAor A2 and undercof B2 find VaXbX con D.

iz? Set?) on B2 to a on A or A2 and under c ofB' find on D.

4. x/ :2 Set the indicating edge N or line Z of the slide to a on scaleE and over 0U B' read V662 on A' or A2.

5. b V202 I? Set the indicating' edge of slide to o on scale E and overbon B2 read V602 on Al or A2.

6. b2 VfL-2 I? Set the indicating edge of slide wto con scale E andabove b on C read b2 V602 on A or A2.

7- (60 by :2. Set o on B to Z1 on A or A2 and at edge of slide read \/(a@3 on scale E.

I? Set b on B2 to c on A or A2 and at edge of slide read on E.

9. Vi :l Set the indicating edge of slide to t on scale E and over a onB2 read Veo on 10. I? Set the indicating edge of slicLe to c on scale Eand under a on B2 read 2/5 on D.

11. :2 Set the indicating edge of slide to c on scale E and under o onB2 read 1* 6/26 on D. y

12. 1 I? Set the indicating edge of slide to o on scale E and over indexon D read It is found in processes of multiplication, division, &c.,that the slide-rule is most convenient, rapid, and acccurate for theengineer, architect, or merchant if graduations be placed on thelogarithmic scales corresponding to fractional parts, such as sixths,eighths, twelfths, &c. Thus when each unit is divided into twelve partson scales C and D the operator is at once enabled to calculate squaremeasure or cubical contents when one or all of the individual lengthsare given in both feet and inches, so the merchant can more easily andreadily calculate the cost of merchandise when the price or length, orboth, are given in fractional parts of their respective units, such asthree and one-eighth cents per yard or ten and one-eighth yards at veand three-eighths cents. Similarly total weights are estimated morerapidly and to a greater nicety when the factor is given in both poundsand ounces. In order to adapt the slide to more varied uses withoutconfusing the scale, it has been found more preferable to divide thescale Dl into graduations of siXths or twelfths and the scale C intograduations of eighths, or vice versa, using diiferent-colored markingsfor them for the sake of clearness. These graduations are clearly shownin Figs. 2 and 3 and are designated by P and S, respectively.

The additional cost of manufacture, the ab sence of the table ofconstants and other useful information usually found on the back of fthe ordinary Mannheim slide-rule, and the impossibility of makingtrigonometrical and other computations without the aid of an additionalslide, together with many inherent disadvantages, lack of adjustabilityagainst warping, &c., are the reasons that the DupleX rule did not findthe favor with calculators that it would seem to merit. Thesedisadvantages become the advantages of my improved rule and will serveto commend it at once to engineers and calculators.

l claim as new and desire to secure by Letters Patente 1.- A slide-rule,consisting of a main piece having two fixed bars, one bar having asingle logarithmic scale the graduated length of the same, and the otherbar having two scales each of half the graduated length of the rule,`

toward the right-hand end of the bar for the graduated length of thesame and the other bar having two scales each progressing from IIO theleft toward the right, one from the left-- hand end to the median pointof the graduated length of the bar and the other from the median pointtoward the right-hand end of' the bar, and -a slide movable between thefixed bars having a logarithmic scale on one side of its longitudinalcenter line in juxtaposition to the single logarithmic scale of one ofthe fixed bars, and two logarithmic scales on the other side of thelongitudinal center line, one progressing from the median point towardthe right-hand end of the slide and the other progressing from themedian point toward the left-hand end of' the slide, substantiallyasdescribed.

3. A slide-rule, consisting of a main piece and two fixed bars, one barhaving a logarithmic scale progressing from the left toward the right ofthe rule for the graduated length of the same and the other bar havingtwo logarithmic scales each progressing from the left toward the right,one from the left-hand end to the median point of the graduated lengthof the bar and the other from the median point toward the right-hand endof the bar, and a slide movable between the two fixed bars having alogarithmic scale on one side of its longitudinal center line injuxtaposition to the single scale of one of the fixed bars, and twologarithmic scales on the other side of' the longitudinal center lineand progressing in opposite directions, one scale progressing in thesame direction as one of the two scales of one of the fixed bars and theother progressing in the opposite direction to the other of' the twoscales, substantially as described.

4. `In a slide-rule, the combination, with a main piece provided withtwo fixed bars, one bar being' provided with two contiguous logarithmicscales, one progressing in one direction from one end of the bar towardits median point and the other scale from said median point in the samedirection to the opposite end of' the bar, the other bar having alogarithmic scale progressing the graduated length of the bar in thesame direction as the scales of' the first bar, of' a slide guidedbetween said fixed bars and provided at the edge adjacent to thelast-named bar with a logarithmic scale extending through the graduatedlength of' the slide and at the opposite edge with two logarithmicscales one progressing in one direction from the median point of theslide to one end of the same, while the other scale progresses in theopposite direction from the median point to the other end of the slide,substantially as described.

5. In aslide-rule, a slide provided along one edge with a logarithmicscale progressing in one direction from one end of the slide to theother and at the opposite side with two logarithmic scales, oneprogressing in one direction .and the other progressing in the oppositedirection, substantially as described.

6. Inaslide-rule, the combination of a main piece consisting of across-piece provided with two fixed bars, each of said fixed bars havinglogarithmic graduations, and logarithmic graduations arranged on thecross-piece representing the cubes of' numbers on one of` the fixed barsand the three-half powers of the numbers on the other fixed bar, and aslide guided between the fixed bars and provided at each edge with arecess, substantially as described. n

7. 'In a slide-rule, the combination of a main piece consisting of' across-piece having logarithmic graduations and provided with two fixedbars, each having` logarithmic graduations, the graduations of' thecross-piece being in cubical relation to the graduations of" one of thefixed bars and a slide having a recess at each end movable between thefixed bars, the edge of which recess serves to place either end of thegraduated length of' the slide in alinement with the graduations on thecrosspiece, substantially as described.

8. In aslide-rule, the combination of a main piece consisting of' across-piece having logarithmic graduations and provided with two fixedbars, each having logarithmic graduations, the graduations of' thecross-piece being in three-half-power relation to the graduations of`one of' the fixed bars, and a slide having a recess at each end movablebetween the fixed bars, the edge of which recess serves to place eitherend of the graduated length of the slide in alinement with thegraduations on the cross-piece, substantially as described.

9. Aslide-rule,comprising amain piece consisting of' a cross-piecehaving demarliations IOO and two fixed bars provided with logarithmicscales, and a slide guided between said fixed bars, said slide beingalso provided with logarithmic scales and having a recess at each end ofthe graduated length of the sa'me, the edges of' which recess serve toplace eitherend of' the graduated length ofthe slide in alinement withthe demarlfations arranged on the crosspiece, substantially asdescribed.

lO. A slide-rule, consisting of a main piece composed of' a cross-pieceand two fixed parallel bars, and a slide guided between the fixed bars,said fixed bars, slide and the cross-piece between the fixed bars beingprovided with logarithmic graduations and each end of' said slide havinga recess, the inner edges of which serve to place it in alinement withthe logarithmic graduations arranged on the crosspiece, substantially asdescribed.

In testimony that I claim the foregoing as my invention I have signed myname in presence of two subscribing witnesses.

LEON VALTER ROSENTHAL.

Witnesses:

GEORGE F. SEvER, HENRY J. SUHRBIER.

IIO

